Solving linear inequalities | Algebra 2

A whole range of answers

The equation $$x + 4 = 10$$ has one solution, $$x = 6$$ . Change the equals sign and the story changes:

x + 4 < 10

Now $$x = 5$$ works, $$x = 0$$ works, $$x = -100$$ works, and so does $$x = 5.99$$ . The solutions are every number less than $$6$$ : a whole stretch of the number line.

Three ways to write a solution set

Inequality	Interval notation	Graph
$$x > 3$$	$(3, \infty)$	open circle at $$3$$ , shade right
$x \le -1$	$(-\infty, -1]$	closed circle at $$-1$$ , shade left
$-2 < x \le 5$	$$(-2, 5]$$	open at $$-2$$ , closed at $$5$$ , shade between

A parenthesis means the endpoint is not included; a bracket means it is. Infinity always gets a parenthesis, because $\infty$ isn't a number you can actually reach.

On a graph, an open circle marks an endpoint that's left out and a closed (filled) circle marks one that's included.

The rules for solving

Solving an inequality works almost exactly like solving an equation. Almost.

Why the flip? Start with something true: $$3 < 7$$ . Multiply both sides by $$-1$$ and you get $$-3$$ and $$-7$$ . On the number line, $$-3$$ is to the right of $$-7$$ , so now $$-3 > -7$$ . Multiplying by a negative mirrors the whole number line, and the order reverses.

When the variable disappears

Sometimes the $$x$$ terms cancel, leaving a statement about numbers only. That statement is either always true or never true.

Practice

Quick check

What is the solution of $-5x \le 20$ ?

Quick check

How do you write $x \ge 2$ in interval notation?

Inequality	Interval notation	Graph
$\(x > 3\)$	$(3, \infty)$	open circle at $\(3\)$ , shade right
$x \le -1$	$(-\infty, -1]$	closed circle at $\(-1\)$ , shade left
$-2 < x \le 5$	$\((-2, 5]\)$	open at $\(-2\)$ , closed at $\(5\)$ , shade between